Important Geometry Formulas with Examples and Rules
Geometry studies points, lines, angles, shapes and their measurements. This chapter covers geometry basics such as angle rules, triangle properties, quadrilateral and polygon formulas, circles, perimeter, area and coordinate geometry. Use the formulas and solved examples to calculate unknown lengths, angles, areas and other quantities in geometry aptitude questions.
What Is Geometry?
Plane geometry concerns two-dimensional figures such as triangles, quadrilaterals, polygons and circles. Solid geometry concerns three-dimensional figures such as cubes, cuboids, cylinders, cones and spheres. Common measurements include length, perimeter, area, surface area and volume. For example, a rectangle with length 8 cm and breadth 5 cm has perimeter 2(8 + 5) = 26 cm and area 8 × 5 = 40 cm².
Geometry Video Lessons
Watch short topic-wise lessons for quick revision.
Polygons: Angle Sum and Exterior Angle
Learn the formulas for the sum of interior angles and exterior angles of polygons, with guidance on applying them to solve geometry problems.
Geometry Formula & Tricks
Important Formulas
Here, l is the length and b is the breadth.
The height must be perpendicular to the selected base.
This applies to a right-angled triangle, where the hypotenuse is opposite the right angle.
Here, r is the radius. Use π = 22/7 or 3.14 as specified.
Here, n is the number of sides. The formula gives the total interior-angle sum.
The slanting side is not used as height unless it is perpendicular to the base.
The dimensions must be expressed in the same unit before calculation.
Quick Tricks
A straight line gives 180°, angles around a point give 360°, and the angles of a triangle give 180°. Apply these fixed sums first to reduce the number of unknowns.
Divide an irregular figure into rectangles, triangles or sectors. Calculate each part separately and add areas, or subtract an unwanted part from a larger known figure.
For sides a, b and c with c as the largest side, the triangle is right-angled if a² + b² = c². This can avoid unnecessary angle calculations.
Geometry Concepts
Lines and Angles
Complementary angles have a sum of 90°, while supplementary angles have a sum of 180°. Vertically opposite angles are equal. When parallel lines are cut by a transversal, corresponding angles are equal, alternate interior angles are equal, and co-interior angles are supplementary.
Triangles and Their Properties
An equilateral triangle has three equal sides and three 60° angles. An isosceles triangle has two equal sides and equal opposite angles. In a right-angled triangle, the side opposite the right angle is the hypotenuse, and a² + b² = c². The exterior angle of a triangle equals the sum of its two opposite interior angles.
Quadrilaterals and Polygons
A quadrilateral has an interior-angle sum of 360°. A parallelogram has opposite sides and opposite angles equal. A rectangle has four right angles, and a square has four equal sides and four right angles. A rhombus has four equal sides, while a trapezium has at least one pair of parallel sides.
Circles
The diameter is twice the radius. The circumference is 2πr and the area is πr². A chord joins two points on a circle, while a tangent touches the circle at one point. The radius drawn to the point of contact is perpendicular to the tangent.
Coordinate Geometry
The midpoint of two points is ((x₁ + x₂)/2, (y₁ + y₂)/2). The slope of a non-vertical line is (y₂ − y₁)/(x₂ − x₁). Points with the same x-coordinate lie on a vertical line, and points with the same y-coordinate lie on a horizontal line.
Geometry Visual Guide & Data
Use the table and chart below to understand the data better.
Common Plane-Geometry Formulas
Perimeter, circumference and area formulas for frequently used two-dimensional figures.
| Figure | Perimeter or circumference | Area |
|---|---|---|
| Square | 4a | a² |
| Rectangle | 2(l + b) | lb |
| Triangle | a + b + c | 1/2 × base × height |
| Parallelogram | 2(a + b) | base × height |
| Circle | 2πr | πr² |
Note: In the square, a is the side. In the rectangle, l and b are length and breadth. In the circle, r is the radius.
Triangle with base and perpendicular height
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Geometry Revision Points
Use these rules and formulas for quick recall.
- Angles on a straight line sum to 180°.
- Angles around a point sum to 360°.
- The interior angles of a triangle sum to 180°.
- The interior-angle sum of an n-sided polygon is (n − 2) × 180°.
- For a right triangle, hypotenuse² = perpendicular² + base².
- Rectangle area = length × breadth; triangle area = 1/2 × base × height.
- Circle circumference = 2πr and circle area = πr².
- Convert all dimensions to the same unit before using a formula.
Geometry FAQs
What is the formula for the area of a triangle?
Area = 1/2 × base × perpendicular height. For base 10 cm and height 6 cm, the area is 1/2 × 10 × 6 = 30 cm².
How is the interior-angle sum of a polygon calculated?
For an n-sided polygon, the sum is (n − 2) × 180°. For an octagon, it is (8 − 2) × 180° = 1080°.
What is the difference between radius and diameter?
The radius is the distance from the centre to the circle, while the diameter passes through the centre from one side to the other. Diameter = 2 × radius.
How do you find the circumference of a circle?
Use circumference = 2πr or πd. For radius 14 cm and π = 22/7, the circumference is 2 × 22/7 × 14 = 88 cm.
When can the Pythagorean theorem be used?
It can be used only in a right-angled triangle. If the perpendicular sides are 5 cm and 12 cm, the hypotenuse is √(5² + 12²) = 13 cm.
What is the area of a parallelogram?
Area = base × perpendicular height. A parallelogram with base 9 cm and height 4 cm has area 36 cm².